Main Page
MediaWiki has been installed.
Consult the User's Guide for information on using the wiki software.
Let's add a link to my blog
$\newcommand{\Re}{\mathrm{Re}\,}
\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
\newcommand{\ket}[1]{\vert #1\range}$
Take a look at this beautifully typesetted equation: \begin{equation} p(n) = \mathcal{C}_n\times {}_1\!F_2\left(n+1;\frac{2n+1}{2}+\frac{\Gamma_\sigma}{\Gamma}, n+\frac{\Gamma_\sigma}{\Gamma}; -\frac{P_\sigma} {2\Gamma} \right)\times \ket{\alpha} \end{equation}
We consider, for various values of $s$, the $n$-dimensional integral \begin{align} \label{def:Wns} W_n (s) &:= \int_{[0, 1]^n} \left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x} \end{align} which occurs in the theory of uniform random walk integrals in the plane, where at each step a unit-step is taken in a random direction. As such, the integral \eqref{def:Wns} expresses the $s$-th moment of the distance to the origin after $n$ steps.
By experimentation and some sketchy arguments we quickly conjectured and strongly believed that, for $k$ a nonnegative integer \begin{align} \label{eq:W3k} W_3(k) &= \Re \, \pFq 32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. \end{align} Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. The reason for \eqref{eq:W3k} was long a mystery, but it will be explained at the end of the paper.
First Section
First Subsection
First subsubsection
<m>x^2</m>
CV file
fa-camera-retro
<a class="list-group-item" href="#"> Home</a> <a class="list-group-item" href="#"> Library</a> <a class="list-group-item" href="#"> Applications</a> <a class="list-group-item" href="#"> Settings</a>
<a class="list-group-item" href="#"> Home</a> <a class="list-group-item" href="#"> Library</a> <a class="list-group-item" href="#"> Applications</a> <a class="list-group-item" href="#"> Settings</a>
- List icons
- can be used
- as bullets
- in lists
...tomorrow we will run faster, stretch out our arms farther...
And then one fine morning— So we beat on, boats against the
current, borne back ceaselessly into the past.